Name
Abstract | ||
|---|---|---|
We introduce an extended class of cardinal L*L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L*L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional ||Ls||L22, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a "smoothness" term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L*L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L*L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this model-based formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1109/TSP.2005.847821 | IEEE Transactions on Signal Processing |
Keywords | Field | DocType |
optimal solution,l smoothing spline,input signal,cardinal l,fast recursive algorithm,standard wiener solution,interpolation constraint,generalized smoothing spline,l-spline signal interpolation problem,privileges solution,wiener filter,exponential spline,optimal discretization,digital filter,gaussian noise,least squares approximation,satisfiability,variational principle,pseudo differential operator,smoothing splines,interpolation,transfer functions,stationary process,least squares estimation,transfer function,minimum mean square error,white gaussian noise,digital filters,smoothing spline,recursive algorithm,digital filtering | Wiener filter,Spline (mathematics),Mathematical optimization,Interpolation,Smoothing spline,Minimum mean square error,Regularization (mathematics),Smoothing,Gaussian noise,Mathematics | Journal |
Volume | Issue | ISSN |
53 | 6 | 1053-587X |
Citations | PageRank | References |
30 | 1.87 | 10 |
Authors | ||
2 | ||